I find that with anything more complex than accidentally bumping into someone, rather than apologizing first thing, I prefer to consider what happened: if I misunderstood, for instance, then what did I mistake for what? I want to straighten it out for myself at the time so that I won't have to worry about making the same mistake over and over.

I prefer to apologize for whatever I actually said or did that turned out not to be a good idea, not for how the other person may have heard it -- though sometimes, "not a good idea" could include overlooking that what I said might trigger them.

If I do fall back on giving a blanket apology without being entirely clear about what I did, I'm probably just saying it to get the other person to back off so that I can think about it some more.